Every now and then there is an Internet meme pointing to health risks from cell phone usage. According to these, carrying an active phone close to your body, or holding the phone to your head while conversing, have some association with cancer, or possibly other physical health risks. Of course the psychological health risks of excessive smart phone usage are well known and discussed, but here I am just concerned with physiological risks to human health. These internet stories were bolstered in 2011 by official reports that additional studies had found possible links between prolonged cell phone usage and certain cancers in humans.
Various professional bodies have done numerous investigations over the decades about this. In particular, the IEEE has looked into it often, since they are closely associated with wireless phone systems, standards, hardware, networks, software, and so on. Most of their studies found no statistically meaningful health effects, although some studies were statistically "inconclusive", and the above mentioned study found "there could be some risk" to humans. The rationale usually offered for minimal risk is that the microwaves used in cell phone networks are "non-ionizing radiation" so cannot break chemical bonds in your body. Without breaking bonds, DNA mutations are impossible, your genes are safe, and so cancers cannot be triggered.
The other potential cell-damage mechanism usually mentioned is that microwaves can heat biological material (like your body), as in a microwave oven. But the microwave power levels used in cell phones (milliwatts) are far too low to cause detectable heating in nearby tissues. With those two potential mechanisms out of the way, it is happily concluded that, aside from that troubling study, there is little risk that cell phone signals can cause cancer or other diseases.
While it is true that microwaves are non-ionizing, the accompanying argument that only thermal effects are important may be too simplistic. Biological molecules may, conceivably, also react to microwaves in additional ways in some cases. Depending on the size, shape and electron charge distribution in the molecules, they may react to microwaves by twisting, vibrating, or rotating (spinning). That is what happens to water molecules in a microwave oven, after all, leading to the desired heating effect. If certain biological molecules happen to resonate with wireless frequencies (which vary over a wide range, from 500 to 5000 MHz or so), then perhaps their response could, in principle, prevent (or speed up) their biological function or behaviour.
Biological molecules have complex behaviours: connecting to or releasing another molecule, being bent, shaped or reshaped inside a cell, passing through cell wall portals, holding two other molecules close together, and so on. If any of these actions is disrupted, sped up, or slowed down in some way, that in turn could interfere with its normal operation inside or between cells. Given the complexity of biochemical reactions, it is probably impossible to say there cannot be any such effect at all. In the worst case, if DNA replication were tweaked even a little, say by increasing the vibrational noise level around the process at some frequency, then copying errors (mutations) might be slightly more frequent. It may therefore, be impossible to rule out an occasional cancer getting a foothold, just at a very low statistical level. Alternatively, if the microwaves disrupt a pore in the cell membrane, it could affect the cell's uptake or discharge of chemicals, such as carcinogens from the cellular environment, or internal wastes. In that case, the microwaves themselves would not "cause" the cancer, but could make the cell more susceptible to cancers caused by those substances.
I do not want to raise fears of cell phones or microwaves in general, and I certainly do not want to start another Internet meme to that effect! However, I have never seen this potential mechanism addressed, much less refuted in what I have read on this subject -- admittedly a small sampling of the available literature. It would be interesting to see what the experts have to say about this. Perhaps they have already examined this possibility and dismissed it? Maybe the microwave frequencies are all wrong for affecting biochemicals? Maybe the resulting vibration or rotational effects are orders of magnitude below the noise level in the cell? I don't know, but it seems that no potential mechanism should be ignored just because it is complicated or difficult to assess.
I note in closing that the same articles reporting negligible health risks of cell phone usage usually also advise that users practice "pragmatic measures" or "prudent precautions" to reduce the (supposedly negligible) risks of microwave exposure. Perhaps after the 2011 reports they are hedging their bets and may now be open to considering the above potential mechanism? Nevertheless, please don't worry about your phone based on my speculations, and do not spread rumours. I am certainly not going to stop using my phone, although I don't often carry it on my person. In any case, there are enough real health hazards in our world to worry excessively about potential ones.
Thursday 22 November 2018
Friday 16 November 2018
The Metaphysics of Mathematics
Mathematics is probably considered the science least associated with or affected by faith or religion. It is all neutral axioms and firm logic, theorems and proofs, so how could beliefs come into it? Nevertheless, there are many metaphysical connections and underlying belief aspects in the field of mathematics. I will provide an example, some further observations, a few questions to ponder, and some references to delve deeper into the subject.
Example Exercise: (using only high-school math)
Let f(x) = 1 for x = rational, and f(x) = 0 for x = irrational
Question: what is the integral between 0 and 1 of f(x)?
If you don't know, an integral is simply the area under the plot of f(x) on an x-y graph between the two limits of x, so there is no need for actual calculus (integration) here.
Depending on your metaphysical assumptions, any of the following answers may be correct:
a) If you were a Pythagorean back in the day, you would not believe that irrational numbers actually exist, so f(x) = 1 for all real numbers and the integral (area) between 0 and 1 would simply be 1.
Even today, there are people who question how real most irrational numbers are since the vast majority of them can never be described exactly, because they are an infinite series of digits in any numerical expansion; e.g. x = 0.876513234923746023700147829432764591034 is not irrational. It can be expressed as a ratio of two integers, and so is rational. Only by continuing such a string of pseudo-random digits to infinity can it be said to be irrational. Thus, a vanishingly small percentage of irrational numbers can be described in any finite form: numbers such as pi, e (the base of natural logarithms), square-root of two, 0.909009000900009000009..., and so on. So if most irrational numbers do not truly exist in any meaningful, presentable sense, one could perhaps say that the rational numbers between 0 and 1 far out weigh the irrational ones, and the integral approaches unity in the limit.
b) However, assuming that all irrational numbers do truly exist (in some sense), it is clear that there are any number of both irrational and rational numbers between 0 and 1. Indeed, we can say that there are an infinity of both type between 0 and 1. We cannot count how many, so we might assume (for now) that for every rational number there is an irrational one. Then half the numbers in that range are rational and half irrational, so that the area is one half (0.5). This might be thought of as a pseudo-average of f(x) over the 0 to 1 range.
c) Not so fast! That is not the full story. According to Georg Cantor (19th century) and most mathematicians since then, the rational numbers are "countable", which means they can be paired one-to-one with the infinite set of integers. He provided a simple demonstration of that. On the other hand, the irrationals cannot be counted in the same way, or at least no one has come up with a scheme for doing so. Thus, officially, the rational numbers are a different degree of infinity than the irrationals.
All infinite "countable" sets of things (numbers or otherwise) are considered to be at the "aleph-zero" level of infinity. The real numbers (rationals plus irrationals) cannot be counted, and are therefore considered as "aleph-one" level of infinity. In an aleph-one set, there are infinitely more items than in an aleph-zero set; not just double infinity, but "infinity squared" so to speak (although infinity is not a number so cannot be squared). If the real numbers are aleph-one, while the rational numbers are aleph-zero, then the irrationals must be aleph-one.
Therefore, rather than being the same number of rationals and irrationals between zero and one, there are infinitely more irrationals. That is, for every rational number between 0 and 1, there are an infinite number of irrationals. In this approach, the integral of f(x) is arbitrarily close to zero; i.e. the vast majority of the point in x have f(x) = 0 and the "average" over x = 0 to 1 is zero. I expect this would be the "official" answer to the question, referred to as the "non-constructivist" view, but it too is not without metaphysical issues.
d) Finally (I hope), the integral operation is often explained as summing up a huge number of narrow dx slices under a plot of the function, between the two limits of x. Thus, for example, when integrating the function g(x) = x, one would take narrow dx slices between x=0 and x=1, and add up their areas, as dx was made smaller and smaller, the integral would converge on a value of one half (0.5) for g(x). The integral calculus allows one to find the limit, i.e. the true area, without adding up the tiny slices. Thus, dx is referred to as an infinitesimal and, in the limit, there are an infinite number of dx slices between zero and one.
This works well for any continuous function like g(x), but f(x) is an "everywhere discontinuous" function. No matter how small you make dx, there will be an infinite number of rational numbers in the slice and infinitely more irrational ones. Thus, there is no way to look at a finite dx, much less an infinitesimal one to determine the "true value" of f(x) in that slice.
The concept of "real infinities" and infinitesimals caused mathematical philosophers a lot of trouble when calculus was being developed back in the 17th century. Still today, infinitesimals are suspect to many philosophers, who basically say we can use calculus because it "works" for functions like g(x), but not because it represents reality or truth. Given such metaphysical doubt, some would conclude that for the function f(x), dx is meaningless, therefore the integral is undefined; i.e. it cannot be computed.
Some Other Observations:
a) Zeno's paradox updated: movement is impossible as it requires something to be in an infinite number of locations in a finite time, and "real infinities" are impossible. Despite seeming to be silly, this paradox still causes metaphysical concerns for some philosophers.
b) Infinity is not a number, but Georg Cantor's transfinite numbers (aleph 0, 1, 2...) tries to make sense of infinite sets of different "sizes", as noted above. Cantor said there is an infinite series of ever increasing infinities (aleph-n), but I have not seen any descriptions above aleph-2, and some mathematicians question how real the higher alephs are. Hence answer (b) above.
c) Kurt Gödel's incompleteness theorems (in the 1930's?): in any mathematical system there are true statements that cannot be proven within that system; i.e. no mathematical system is "closed". Some people (e.g. John Lucas and Roger Penrose) claim this means that humans are not computers.
d) Euler's identity: 1+ e^i*pi = 0
This combines five, seemingly unrelated fundamental mathematical objects in a relation: zero, unity, two transcendental numbers, even an "imaginary number". How weirdly beautiful is that?
e) Fractals are "fractional dimension" mathematical objects, infinite complexity arising from simplicity, deep beauty from simple equations. They have real-life applications: e.g. the shape of fern fronds, and how long is the coastline of England? (depends on what scale you use).
f) Humorous theorem: there is no such thing as a boring number!
Proof: if there were boring numbers, then the set of boring numbers would have a first member, "the first boring number", and that would be very interesting indeed, so it could not be boring.
Questions to Ponder (based on your own metaphysics):
1. Are mathematical objects (e.g. polygons, relations, theorems) invented or discovered?
2. Do mathematical objects (e.g. numbers, geometric results) really exist?
They are not matter, energy, or information! (this evokes Plato's metaphysical “forms”).
3. If mathematical objects exist, are they part of the universe? If so, could math be different in
another universe or creation? i.e. could God have created mathematics differently?
4. Why is mathematics so effective for science? Do math-based physical “laws” actually operate
the Universe or are they just human models?
References for Further Study:
How religion and faith (metaphysics) underpin mathematics:
https://frame-poythress.org/a-biblical-view-of-mathematics/
A brief Reformed Christian view of mathematics:
https://www.redeemer.ca/resound/stewarding-talent-in-mathematics/
A quite different (contrary) view:
http://steve-patterson.com/the-metaphysics-of-mathematics-against-platonism/
A brief but thoughtful book review on the subject:
https://reviewcanada.ca/magazine/2014/09/the-metaphysics-of-math/
If you want to get lost in the subject! (quickly gets deep):
https://plato.stanford.edu/entries/philosophy-mathematics/
This last reference shows that the subject remains an active and serious area of professional inquiry and debate.
Example Exercise: (using only high-school math)
Let f(x) = 1 for x = rational, and f(x) = 0 for x = irrational
Question: what is the integral between 0 and 1 of f(x)?
If you don't know, an integral is simply the area under the plot of f(x) on an x-y graph between the two limits of x, so there is no need for actual calculus (integration) here.
Depending on your metaphysical assumptions, any of the following answers may be correct:
a) If you were a Pythagorean back in the day, you would not believe that irrational numbers actually exist, so f(x) = 1 for all real numbers and the integral (area) between 0 and 1 would simply be 1.
Even today, there are people who question how real most irrational numbers are since the vast majority of them can never be described exactly, because they are an infinite series of digits in any numerical expansion; e.g. x = 0.876513234923746023700147829432764591034 is not irrational. It can be expressed as a ratio of two integers, and so is rational. Only by continuing such a string of pseudo-random digits to infinity can it be said to be irrational. Thus, a vanishingly small percentage of irrational numbers can be described in any finite form: numbers such as pi, e (the base of natural logarithms), square-root of two, 0.909009000900009000009..., and so on. So if most irrational numbers do not truly exist in any meaningful, presentable sense, one could perhaps say that the rational numbers between 0 and 1 far out weigh the irrational ones, and the integral approaches unity in the limit.
b) However, assuming that all irrational numbers do truly exist (in some sense), it is clear that there are any number of both irrational and rational numbers between 0 and 1. Indeed, we can say that there are an infinity of both type between 0 and 1. We cannot count how many, so we might assume (for now) that for every rational number there is an irrational one. Then half the numbers in that range are rational and half irrational, so that the area is one half (0.5). This might be thought of as a pseudo-average of f(x) over the 0 to 1 range.
c) Not so fast! That is not the full story. According to Georg Cantor (19th century) and most mathematicians since then, the rational numbers are "countable", which means they can be paired one-to-one with the infinite set of integers. He provided a simple demonstration of that. On the other hand, the irrationals cannot be counted in the same way, or at least no one has come up with a scheme for doing so. Thus, officially, the rational numbers are a different degree of infinity than the irrationals.
All infinite "countable" sets of things (numbers or otherwise) are considered to be at the "aleph-zero" level of infinity. The real numbers (rationals plus irrationals) cannot be counted, and are therefore considered as "aleph-one" level of infinity. In an aleph-one set, there are infinitely more items than in an aleph-zero set; not just double infinity, but "infinity squared" so to speak (although infinity is not a number so cannot be squared). If the real numbers are aleph-one, while the rational numbers are aleph-zero, then the irrationals must be aleph-one.
Therefore, rather than being the same number of rationals and irrationals between zero and one, there are infinitely more irrationals. That is, for every rational number between 0 and 1, there are an infinite number of irrationals. In this approach, the integral of f(x) is arbitrarily close to zero; i.e. the vast majority of the point in x have f(x) = 0 and the "average" over x = 0 to 1 is zero. I expect this would be the "official" answer to the question, referred to as the "non-constructivist" view, but it too is not without metaphysical issues.
d) Finally (I hope), the integral operation is often explained as summing up a huge number of narrow dx slices under a plot of the function, between the two limits of x. Thus, for example, when integrating the function g(x) = x, one would take narrow dx slices between x=0 and x=1, and add up their areas, as dx was made smaller and smaller, the integral would converge on a value of one half (0.5) for g(x). The integral calculus allows one to find the limit, i.e. the true area, without adding up the tiny slices. Thus, dx is referred to as an infinitesimal and, in the limit, there are an infinite number of dx slices between zero and one.
This works well for any continuous function like g(x), but f(x) is an "everywhere discontinuous" function. No matter how small you make dx, there will be an infinite number of rational numbers in the slice and infinitely more irrational ones. Thus, there is no way to look at a finite dx, much less an infinitesimal one to determine the "true value" of f(x) in that slice.
The concept of "real infinities" and infinitesimals caused mathematical philosophers a lot of trouble when calculus was being developed back in the 17th century. Still today, infinitesimals are suspect to many philosophers, who basically say we can use calculus because it "works" for functions like g(x), but not because it represents reality or truth. Given such metaphysical doubt, some would conclude that for the function f(x), dx is meaningless, therefore the integral is undefined; i.e. it cannot be computed.
Some Other Observations:
a) Zeno's paradox updated: movement is impossible as it requires something to be in an infinite number of locations in a finite time, and "real infinities" are impossible. Despite seeming to be silly, this paradox still causes metaphysical concerns for some philosophers.
b) Infinity is not a number, but Georg Cantor's transfinite numbers (aleph 0, 1, 2...) tries to make sense of infinite sets of different "sizes", as noted above. Cantor said there is an infinite series of ever increasing infinities (aleph-n), but I have not seen any descriptions above aleph-2, and some mathematicians question how real the higher alephs are. Hence answer (b) above.
c) Kurt Gödel's incompleteness theorems (in the 1930's?): in any mathematical system there are true statements that cannot be proven within that system; i.e. no mathematical system is "closed". Some people (e.g. John Lucas and Roger Penrose) claim this means that humans are not computers.
d) Euler's identity: 1+ e^i*pi = 0
This combines five, seemingly unrelated fundamental mathematical objects in a relation: zero, unity, two transcendental numbers, even an "imaginary number". How weirdly beautiful is that?
e) Fractals are "fractional dimension" mathematical objects, infinite complexity arising from simplicity, deep beauty from simple equations. They have real-life applications: e.g. the shape of fern fronds, and how long is the coastline of England? (depends on what scale you use).
f) Humorous theorem: there is no such thing as a boring number!
Proof: if there were boring numbers, then the set of boring numbers would have a first member, "the first boring number", and that would be very interesting indeed, so it could not be boring.
Questions to Ponder (based on your own metaphysics):
1. Are mathematical objects (e.g. polygons, relations, theorems) invented or discovered?
2. Do mathematical objects (e.g. numbers, geometric results) really exist?
They are not matter, energy, or information! (this evokes Plato's metaphysical “forms”).
3. If mathematical objects exist, are they part of the universe? If so, could math be different in
another universe or creation? i.e. could God have created mathematics differently?
4. Why is mathematics so effective for science? Do math-based physical “laws” actually operate
the Universe or are they just human models?
References for Further Study:
How religion and faith (metaphysics) underpin mathematics:
https://frame-poythress.org/a-biblical-view-of-mathematics/
A brief Reformed Christian view of mathematics:
https://www.redeemer.ca/resound/stewarding-talent-in-mathematics/
A quite different (contrary) view:
http://steve-patterson.com/the-metaphysics-of-mathematics-against-platonism/
A brief but thoughtful book review on the subject:
https://reviewcanada.ca/magazine/2014/09/the-metaphysics-of-math/
If you want to get lost in the subject! (quickly gets deep):
https://plato.stanford.edu/entries/philosophy-mathematics/
This last reference shows that the subject remains an active and serious area of professional inquiry and debate.
Saturday 10 November 2018
Published a Book Have I
Yes, it's true, I am now a published author! After many years of soul searching and prayer, and not a little procrastination, I finally succumbed to the subconscious nagging and wrote up my book on the bioethics of abortion: Finding the Balance: quantitative ethics resolves the abortion issue. After some positive feedback from reviewers and subsequent revisions, along with the usual battles with my word processor for control of the formatting, I actually got it published! It is a simple e-book, available on Amazon at: https://www.amazon.ca/dp/B07JM8L4RT/
The Kindle app is free of charge for computers, tablets and smart phones, so you can read it on almost any device. I also wanted to make the book free to download, but Amazon would not let me offer it for less than $0.99. I won't copy the book description here as you will see that when you open the link. It is a short book and (I think) an easy read, even with its graphs and yes, equations. The first three pages are viewable free on-line. They give my purpose for writing the book and begin to introduce the topic.
Those of you who know me well may think you already know the gist of what I've written, but you would be mistaken. My "quantitative ethics" approach was developed to be as completely neutral as possible; that is, to avoid or counter all the usual emotional arguments on both sides of the abortion debate, as impossible as that may seem. The analysis and results I arrive at depart from a purely pro-life position -- hence the soul searching on my part alluded to above. I will not reveal the conclusions of my efforts here, but I hope my book will help reframe the abortion debate along objective, defendable lines. The quantitative ethics approach is also offered and applied as a new tool for use in other bioethical subjects, such as end-of-life issues.
I expect that most reviews (if I get any at all) will be negative. After all, most people who read books about abortion will likely already be either solidly pro-life or pro-choice, and they will not like what I have written! This could be a good thing as long as it doesn't dissuade everyone else from reading it. The book is mostly aimed at the majority of people, neither pro-choice nor pro-life (or perhaps both), who may be looking for a reasonable resolution to the issue, one that eschews both extremes.
I hope your curiosity will be piqued enough to risk the minimal cost and find out what I have come up with in this book. And if you indeed read it, please consider posting a review on Amazon. Good or bad, any publicity is better than none!
The Kindle app is free of charge for computers, tablets and smart phones, so you can read it on almost any device. I also wanted to make the book free to download, but Amazon would not let me offer it for less than $0.99. I won't copy the book description here as you will see that when you open the link. It is a short book and (I think) an easy read, even with its graphs and yes, equations. The first three pages are viewable free on-line. They give my purpose for writing the book and begin to introduce the topic.
Those of you who know me well may think you already know the gist of what I've written, but you would be mistaken. My "quantitative ethics" approach was developed to be as completely neutral as possible; that is, to avoid or counter all the usual emotional arguments on both sides of the abortion debate, as impossible as that may seem. The analysis and results I arrive at depart from a purely pro-life position -- hence the soul searching on my part alluded to above. I will not reveal the conclusions of my efforts here, but I hope my book will help reframe the abortion debate along objective, defendable lines. The quantitative ethics approach is also offered and applied as a new tool for use in other bioethical subjects, such as end-of-life issues.
I expect that most reviews (if I get any at all) will be negative. After all, most people who read books about abortion will likely already be either solidly pro-life or pro-choice, and they will not like what I have written! This could be a good thing as long as it doesn't dissuade everyone else from reading it. The book is mostly aimed at the majority of people, neither pro-choice nor pro-life (or perhaps both), who may be looking for a reasonable resolution to the issue, one that eschews both extremes.
I hope your curiosity will be piqued enough to risk the minimal cost and find out what I have come up with in this book. And if you indeed read it, please consider posting a review on Amazon. Good or bad, any publicity is better than none!
Thursday 8 November 2018
Now Where Was I?
After a four (4) year hiatus (Four Years - really? That's forever in the blogosphere!), I am returning to capture additional THoughts, OPinions and IDeas here at my blog. Why did I stop posting? Not sure; I guess I was preoccupied with other crucial aspects of retired living: cycling, moving house, grandchildren, reading, writing, volunteering, watching Netflix, procrastinating, etc. Or maybe, having temporarily run out of ideas, I stopped and just didn't get back to it.
Why am I returning now? Well the THOPIDs have been bubbling up in recent months, demanding to escape from my head and be captured in articulate English, so I have been making notes and have now gathered enough to push me over the threshold of starting again. And who knows, maybe this time I'll actually tell people about my blog, and someone may even read it. 😉
In any case, please continue to ignore the dates on my entries, especially the older ones. None of them are particularly time limited, or tied to past news and events. And having read them over again recently, I still hold pretty much all the same positions. I'll let readers decide whether that makes me dependable and consistent, or just stuck in my ways. Perhaps once captured, the thoughts are forever carved in the cloud and don't want to be changed? Or maybe, having written them out, they vacate my mind. After all, why expend wetware resources when we have the cloud?
All this to say, welcome to my blog, or welcome back if you happened to stumble across it earlier. I also welcome any comments you may have, but reserve the right to delete any that are nasty or abusive.
Now what was that I wanted to write about?...
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